Analytical solutions to VBI system for non-simply supported boundary conditions

Matlab codes developed to calculate vehicle and bridge responses (displacement, velocity, and acceleration) for a theoretical vehicle bridge interaction (VBI) system for non-simply supported boundary conditions including both ends fixed, fixed simply supported, and one end fixed the other end free (cantilever) boundary condition. Both the vehicle and bridge damping effects and multiple bridge vibration modes are considered. Hypothesis: Multiple bridge dynamic information may be extracted from the vehicle that travels on it. Assumption: 1. The magnitude of the vehicle acceleration signal (gravitational direction) is negligible compared to the gravitational acceleration constant (g), say <10%. Analytical solutions will be invalid if this condition is not reasonably met, say >20%; 2. Uniformly distributed bridge property (mass, damping, section stiffness); 3. Vehicle travelling speed is constant. Limit: 1. Based on Bernoulli-Euler beam theory, flexure effects caused by shear forces, rotary inertial forces, and axial forces are not considered; 2. For zero initial conditions of bridge and vehicle only; 3. Other flaws may also apply, use with caution. Note: Two forms of solutions are presented for cross-checking purposes, the numerical results for vehicle and bridge are almost the same, only allow for minor differences due to truncation issue existed in numerical calculations.本研究开发了Matlab代码，用于计算非简支边界条件下理论车辆-桥梁耦合（Vehicle Bridge Interaction, VBI）系统的车辆与桥梁响应（位移、速度与加速度），其中非简支边界涵盖两端固定、固支简支以及一端固定一端自由（悬臂）边界工况。研究同时纳入车辆与桥梁的阻尼效应，以及桥梁的多阶振动模态。 研究假说：可通过行驶于桥梁的车辆提取桥梁的多维度动态信息。 前提假设：1. 车辆沿重力方向的加速度信号幅值相较于重力加速度常数(g)可忽略不计，例如小于10%；若该条件未合理满足（例如大于20%），则解析解将失效；2. 桥梁的质量、阻尼与截面刚度属性均呈均匀分布；3. 车辆行驶速度恒定不变。 研究限制：1. 本研究基于伯努利-欧拉（Bernoulli-Euler）梁理论，未考虑剪力、转动惯量与轴力引发的弯曲效应；2. 仅适用于桥梁与车辆的零初始条件场景；3. 可能存在其他未被涵盖的缺陷，使用时需谨慎。 注意事项：为实现交叉验证，本文提供了两种形式的解，车辆与桥梁的数值计算结果几乎一致，仅因数值计算中存在的截断误差存在微小差异。